Hello,
I am looking for a source that discusses and teaches hyperbolic geometry from a synthetic approach (As opposed to the common analytinc approach in the poincare disk). I am looking for something more in spirit with eucld's elements or hilbert's geometry book.

I'll look into them all, but of those three, which would you recommend as the 'best'?
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BladeOct 10 '12 at 0:42

Depends for what. Carlslaw is the most illustrated and elementary, then Sommerville (which has exercises, and on p. 27 refers "the reader who wishes to study the development of non-euclidean geometry from a set of axioms" to Coolidge). Coolidge is more dry and more complete, with e.g. several chapters (IX, X, XVI) on line geometry in hyperbolic space: complexes, congruences, Malus-Dupin theorem, etc.
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Francois ZieglerOct 10 '12 at 2:32